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An Analysis of Mortgage Termination Risks: A Shared Frailty Approach with MSA-Level Random Effects

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Abstract

Investigating the residential mortgage defaults and prepayments has been the subject of research for the past three decades. The literature on mortgage default and prepayment is often used to inform credit risk policies and asset pricing strategies. This literature has evolved from the use of logistic regressions to the use of survival and frailty models that control for unobserved heterogeneity. In this paper, we apply a shared-frailty survival model to analyze the mortgage termination risks. In particular, we investigate whether mortgages originated in the same Metropolitan Statistical Area (MSA) share common unobserved factors and how these factors affect the mortgage termination risks. The paper demonstrates that MSA-level frailty, together with other risk factors, has significant effects on the probability of mortgage terminations risks.

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Notes

  1. The doubly-stochastic property assumes that both the arrival of the termination and the hazard rate of the termination risk are random.

  2. The gamma density function of a random variable x is given by \( f\left( {x;a,\gamma } \right) = x^{{\alpha - 1}} \frac{{e^{{ - {x \mathord{\left/ {\vphantom {x \gamma }} \right. } \gamma }}} }}{{\gamma^{\alpha } \Gamma \left( \alpha \right)}} \) for \( x > 0,\alpha > 0,\gamma > 0 \). The expected value of x is \( E\left( x \right) = \alpha \gamma \) and variance is \( Var\left( x \right) = \alpha \gamma^2 \). The parameter α is referred as the scale parameter and the γ as the shape parameter.

  3. The estimation of the shared-frailty model in STATA consists of two steps. In the first step, the optimization is in terms of θ only. For fixed θ, the second step consists of fitting a standard Cox model via penalized log likelihood, with v ki introduced as estimable coefficients of dummy variables identifying the group. For more details of estimation, please see the Survival Analysis and Epidemiological Tables Reference Manual (2007).

  4. The FICO score is available after 1998. However, not every mortgage originated after 1998 was documented with its borrower’s FICO score. In the dataset, around 73 percent of the mortgages originated after 1998 were attached with borrower’s initial FICO scores at the origination.

  5. The spread is calculated as: \( {{\left( {r_c - r} \right)} \mathord{\left/ {\vphantom {{\left( {r_c - r} \right)} {r_c \times 100}}} \right. } {r_c \times 100}} \), where r c is the contract rate and r is the proxy of the market interest rate.

  6. The HPI is a weighted, repeated sales-index, meaning that it measures average price changes in repeat sales or refinancing on the same properties.

  7. The racial composition and median income for neighborhoods are relatively stable. We merged subsample of mortgage (which were originated after 1998) data with the Bureau Census 2000 survey data. Thus, for each mortgage which was originated after 1998, each one has ZIP code-level information of racial composition and median income characteristics.

  8. The subsample estimation of the prepayment risk is not robust in terms that the sign of the coefficient of SPREAD has been changed. However, data for mortgages originated after 1998 can still be viewed as a valid test since the signs of the FICO score and the demographic variables are consistent for all three regressions.

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Kau, J.B., Keenan, D.C. & Li, X. An Analysis of Mortgage Termination Risks: A Shared Frailty Approach with MSA-Level Random Effects. J Real Estate Finan Econ 42, 51–67 (2011). https://doi.org/10.1007/s11146-009-9179-x

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